On bijections of Lorentz manifolds,which leave the class of spacelike paths invariant

Suppose that (M, g) and (M′, g′) are Lorentz manifolds, and that f: M → M′ is a bijection, such that f and f^-1 preserve spacelike paths (f: M → M′ has this property, if for any spacelike path γ: J → M in (M ,g), the composition fγ: J → M′ is a spacelike path in (M′, g′)). Then f is a (manifold-) homeomorphism.This statement is the ‘spacelike’ version of an analogous ‘timelike’ theorem (Hawking, King and McCarthy 6] and Göbel 2] for strongly causal, and Malament 10] for general Lorentz manifolds).With this result it is possible to prove a conjecture of Göbel 3] which states that every bijection between time-orientable n-dimensional (n ? 3) Lorentz manifolds which preserves spacelike paths is a conformal C^∞-diffeomorphism.